Subject: Re: [HM] What is good math history?
From: Jan Mycielski (jmyciel@euclid.Colorado.EDU)
Date: Thu Jan 06 2000 - 23:27:00 EST
Dear HM Readers,
As the author of two thoughts that once appeared on HM (see Julio
Gonzalez Cabillon, HM, 30 Sep 1998), and were later discussed and
criticised, I feel that it is time to reiterate those thoughts with
some additional comments. The two thoughts (now slightly modified) are the
following.
(1) Good history of mathematics is the history of good ideas (in
particular it ought to give proper credit and appreciation of the true
inventors of the good ideas, whenever possible).
[For example since I am a novice to history of math. I do not know
if Descartes should be credited with the invention of cartesian
coordinates. I heard that it was Fermat who really did it. Or was it both?
Surely calculus textbooks should tell us such things (and give references),
but those which I saw did not give it. After all this idea is even more
basic than calculus itself. I will be indebted for this information.
For example I am already indebted to Moore for the information
that the axiom of replacement of ZFC was (essentially) proposed by Cantor
himself, who wrote (in a letter) that a collection equipollent to a set
must be also a set. It is easy to check that this in the presence of the
comprehension axiom and other axioms (which Cantor often used) yields
replacement.]
(2) Good history should allow us to distinguish real contributions
from error or fraud. It should explain how stagnation arises and what are
its causes.
[There exist striking examples of long periods of stagnation
or regress in the growth of mathematical (and scientific) knowledge in
some cultures, and it is bad history to focus on attenuating events (like
tiny progress in spite of all), covering up the reality of those bad
periods, or ignoring the causes of those unfortunate phenomena.]
I wish to thank Eric Schechter for writing the letter attached
below which seems to be the first in a long sequence which treats my (1)
and (2) with understanding (and clarity). And, I would like to add a few
more comments. {For it is always easy to criticise any pronouncement
(say mine) if one does not try first to interpret it in a constructive way
and one chooses instead an unintended interpretation which is wrong.
[An example: the criticism offered by Gilbert Ryle of Descartes' theory
of the mind by interpreting it as a mix up of categories of concepts. This
is bad history because Descartes lived in a time when neither neither the
complexity of the microstructure of the brain nor the concept and
simplicity of universal Turing machines was known or could have been
suspected. For those reasons it was entirely natural for Descartes to
think that we are moved by a nonphysical soul, and this idea was not due
to any logical error. And the book of Ryle ignores completely the relevance
of modern anatomy, physiology and automata theory for the problem of
explaining the mind.]
So let me return to the question what is interesting history, and
propose the following answer.
History is interesting if it interests intelligent broad minded
people. [But it may take a long time to get appreciated, and it is not
made interesting by the fact that it is payed for (even very well payed
for). In fact inbred scholarship, erroneous pedagogy, and even plain
sharlatanry are often well payed, and history is vulnerable to these
evils.]
The intended effect of the above definition of "interesting
history" is to convince the reader that his personal taste is important
and not to be lightly treated or sacrificed on the altar of duty (as
defined by others). In other words the evaluation is in his hands;
in this matter respect for one's own instincts is essential.
Now, I agree with my critics that (1) and (2) may suggest that the
history of bad ideas or prejudice is not interesting at all. This is going
too far. Indeed history of bad ideas is somewhat like paleontology, and
paleontology is interesting and even fascinating inspite of the fact that
id deals with extinct species. However, I feel that failed human ideas are
MUCH LESS interesting than dinosaurs and other extinct species (specialists
may disagree, but they should not forget too often about the existence of
non-specialists).
Be it as it may, there are the questions of priority which history
should resolve. Many mathematicians feel that they should give proper credit
to the authors of good ideas, and correct the confusion (if any) between
the authors of those ideas and their followers or those who were less wise
or less lucky (even if more eloquent or lauder.)
For example it would be the role of historians to make mathematicians
and philosophers more aware of the fact that treating the history of Logic
and Foundations of Mathematics along the lines of the traditional division
into logicism, formalism and intuitionism is misleading. Indeed it
emphasises unreasonably the role of failed ideas while it hides the history
and the importance of the good ideas and diminishes the appreciation of
their authors.
[Here is my own attempt to do some history:
1. Logicism is a failed idea (it is the impossible program of
deriving mathematics from logic; impossible if one distinguishes logic
from set theory. The term logicism percolated from a time when the
simple theory of types was regarded as a part of logic. Today it is
classified more commonly as a weak set theory. We know that it is too
weak to develop mathematics in it. The standard system for developing
mathematics is the axiomatic set theory ZFC.)
2. Formalism: the claim that mathematics is an arbitrary game of
symbols. It is a misrepresentation of the ideas of Poincare, Skolem and
Hilbert. This misrepresentation seems to have been invented by the
Platonists and the intuitionists (Brouwer) in order to popularise their
criticism of the constructive view of mathematics due to Poincare Skolem
and Hilbert (the good idea). This good idea should be called something
else (perhaps rationalism plain and simple), however it is not altogether
trivial. To understand it fully one must appreciate Hilbert's epsilon
formalism (which he published in 1923). And one must appreciate the fact
that this formalism eliminates the need for Platonic assumptions in the
ontology of mathematics.
3. Intuitionism is a rather unsuccessful attempt to develop
mathematics by means of a truncated logic. This attempt was never clearly
justified on philosophical (or any other) grounds, and it did not
generate any superior kind of mathematics. In fact there is a defect in
the practice of intuitionists. Their proofs yield more information than
they state in their theorems. Thus they hide their results in their proofs
unlike classical mathematicians who try to express as much as possible or
reasonable in their theorems.
{However, a historian cannot miss the fact that not only
rationalism (as defined above) but also Platonism remains a very popular
idea among mathematicians. Although I have not heard anyone (other than
myself) stressing the difference between imaginary objects and real
objects.}
Thus you can see why it is bad history to treat foundations of
mathematics as a tree with three branches (the first branch is dead
the second is a misinterpretation and the third is somewhat sectarian),
while the inventors of the living ideas are Frege (Logic), Cantor &
Zermelo (Set Theory) Skolem and Hilbert (again Logic).
I think that good historians can rectify the confused state of
the culture of mathematicians and philosophers on this point.
Another example: The emphasis on teaching axiomatic geometric
theories (e.g. Euclidean axiomatic geometry), as opposed to studies of
various geometric structures (models).
This would seem reasonable in the times of Newton, or even until
Zermelo's 1908 paper. But why should it continue till today? I believe
that it stems from deficient education among geometers and historians.
Since they are not know Logic and Foundations of Mathematics well enough
they are not aware of the fact that the theory of real closed fields is
essentially the only axiomatic theory below "second order" arithmetic
(analysis) which is really alive today. It is quite unvieldy axiomatise
the required transcendental functions which are needed if one wants to go
above elementary geometry without introducing the axioms of analysis. Thus
geometry has lost the foundational role which it held since Euclid till
Newton. Today geometry is akin to algebraic theories studying various
classes of models (e.g. the class of simple groups in algebra, or the
sphere and the hyperbolic plane in geometry). Axiomatic definitions of
sufficiently rich theories of such models are tedious and are neither
useful nor inspiring. In this area, it is not the axioms but the models
that are the objects of living studies and full analysis should be placed
at our disposal.
Thus I believe that only good history can teach this change in the
organisation of mathematics which happened at the beginning of this
century, and to teach mathematicians how ultimate precision and definitive
foundations of mathematics (ZFC) were achieved and how a subsequent idea
of Hilbert (the epsilon formalism) solved the problem of the ontology of
pure mathematics removing the apparent need of Platonic assumptions. The
historian must find the golden threads in the rich carpet of human
efforts, and not pretend that all historical facts are equally important
or interesting. Many schools and ages produced little or nothing
substantial, and it is interesting why it was so. Those fact should be
studied (for we do not want to repeat those errors).
[Of course I have my suspicions; one can blame stagnation or
regress of knowledge on lack of resources I mean general poverty (to some
extent) and on bad authorities (to a much greater extent). Indeed it is
improbable that there were ages (in large nations) which did not produce
talented people. But they were stopped, indeed the temptation of
authorities (in politics, in culture, or in religion) to rule the ideas of
others must be strong (at least we see it time and again in the history of
humanity). And it seems to me that this obstacle (lack of intellectual
freedom in a particular area of knowledge, or the rule of ineducated
authorities) has not been sufficiently studied, however this seems to be
the cause of some of the largest disasters in the history of civilisations.
This is one of the reasons for which giving proper credit to the
inventors of important ideas (or doing good history) is so important and
is the sign of health of civilisations.]
Next semester I will teach history of math. One of the main threads
I intend to develop is the history of the real number system. (I see a
progression: integers, rationals, constructible nos., algebraic nos.,
computable nos., ordinal-definable nos., the continuity of the real line,
etc. That should keep us busy for a while, and a lot of great ideas are
involved. Most of them easy to present or to recall and to attach to
concrete people and ages. Even the history of calculus can be woven in
this topic, at least as a motivating topic.)
It seems to me that Bourbaki's historical fascicles were a very
commendable effort to build an interesting history of mathematics. (It is
unfortunate that the Bourbakists were not sufficiently interested in
foundations of mathematics to lift us all to a reasonable level in this
area.)
Regards to all
Jan Mycielski
P.S. I am indebted to J. J. Castillos for a letter in HM (Nov. 12 1999),
and to Eric Schechter for the letter attached below, which stimulated the
writing of the above.
On Wed, 11 Nov 1998, Eric Schechter wrote:
>
> Some of these questions have two meanings.
>
> I'm a relatively new member of this list, so forgive me if I'm stating
> what is already obvious. It is becoming apparent to me that we have at
> least two substantially different groups of people on this list, who are
> thinking about how to achieve different objectives --
>
> Some of the people in this list are very learned historians of math, in
> many cases engaged in writing books or research papers about the subject.
>
> Others (like me) are newcomers to the subject -- I'm presently teaching
> History of Math for my first time, because the fellow who used to teach
> it here has just retired.
>
> So the question "What should we do in the history of math" really refers
> to at least two different things: (i) What should we do in our research
> on that subject, and (ii) What should we do in that subject when we teach
> it to undergraduates? There may be other meanings too.
>
> : Did Grattan-Guinness mean we just stick to the records of an
> : epoch when we are describing mathematical work of that epoch,
> : and not be guided by a current state of mathematics, as we see
> : it? If the latter, we are faced with vexing historiographical
> : questions about how we can sufficiently eliminate relevant
> : parts of ourselves and our cultures from our historical
> : interpretations,
>
> It is my experience that my undergraduate students have very little
> knowledge of the mathematics of our own modern culture. I find that in
> order to make any sense out of the history, I first have to teach them
> some of the more recent stuff. For instance, in order to explain what
> infinitesimals are and what Dedekind's definition of the reals meant
> and why angles can't be trisected and several other historical topics,
> I first have to give a modern definition of a "field." I suppose things
> could be taught in chronological order, but I think it would take much
> much longer that way. The modern, abstract definition gives us a
> framework in which we can talk about all these older topics, skipping
> over the old confusions and misunderstandings -- or perhaps discussing
> some of the confusions and misunderstandings *after* we've considered
> the correct explanations.
>
> I guess this makes me a "Whig" historian. Can someone recommend one or
> two elementary things I should read, to begin to reduce my Whigginess?
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